1. Introduction

During this lab a beam was tested in order to find the relationships between load, bending moment, stress and strain, slope and the deflection in a cantilever beam which was the main objective. The main purpose was to understand the fundamental principles that have to be taken into account before designing and manufacturing a beam or using one as part of a design.

2. Theory

The theory behind this lab can be categorized to 2 different topics, bending moment and stress being the first, the second being slope and deflection. Each one is discussed below:

2.1 Bending Moment and Stresses

Bending moment is a moment produced by a load applied on a surface that causes it to bend. In the case of the lab the surface is the beam and with the applied at the end of it. In order to calculate the bending moment it is necessary to draw a Free Body Diagram indicating all the forces applied upon the beam. Picture 1 is the free body diagram of the beam. F is the load applied, R is the reaction from the clamps (the bar is fixed on the bench by two G clamps) and M is the bending moment. Since we need to have equilibrium and no forces on the x-axis there are two equations we should use the following equations: If we resolve the vertical forces for the whole beam

ΣFY=0↔F-R=0 ↔

F=R (1) By taking moments anti-clockwise about right hand side for the whole beam WL+ M=0 ↔

M=-WL (2)

Equation (2) is the bending moment of the beam (measured in Nm), while the equation (1) describes the sheer force. Picture [ 2 ]

For the bending stress in the beam it is helpful to know the second moment of area (I) first. The shape of the bar is rectangular as indicated on Figure

Iz=y2dA=-d2d2y2 bdy=by33-d2d2=bd324+bd324↔

Iz=bd312 (3)

The Second moment of area is measured in mm4

The bending stress is σx=εxE where εx is the strain and E is the Young’s modulus. (The stress is measured in N/m2) There is a last equation that connects bending stress with bending moment. MI=σxy=ER|

2.2 Deflection and Slopes

In order to find the equations for slope and deflection it is necessary to use formula (1) and (2). It is known that M =-EId2vdx2. Thus the bending equation can be rewritten as -EId2vdx2=-Wx (x is a random length of the beam always smaller than the whole length L). If it is integrated once we will get the slope equation

-EIdvdx=-W2x2+C1 (4) If we integrate again (4) then we have deflection equation -EI=-W6 x3+C1x+C2 (5)

By inserting boundary conditions to determine the constants of integration the final equations for the beam slope and beam deflection can be determined. At x=L: dv/dx=0 and v=0

4↔0=-W2 L2+C1↔

C1=WL22

5↔0=-W6 L3+WL22 L+C2↔

C2=-WL33

Hence 4 and 5 can be re-written as

Beam Slope: dvdx=W2EI(x2-L2) [m]

Beam Deflection: v=W6EI L-x2(2L+x) [m]

3. Equipment

The equipment used during the lab session is the following:

1) Bench

Used to place the equipment

2) Two G-Clamps

Placed parallel to each other to hold firmly the beam in place

3) One Aluminum Cantilever Beam

The main subject of the experiment, with strain gauges attached both on top and bottom. The Young’s Modulus of the beam is 70GPa. It was marked at 50mm intervals to facilitate the procedure.

Picture 3

4) Strain gauge bridge amplifier with digital readout

The strain amplifier provides the readings of the difference in strain between the top and bottom surfaces of the beam (measured in micro strains).

5) One hanger with weights (10 blocks of 100 grams each). It is used as the load that is applied at the end of the cantilever bar. The hanger helps in applying the load (there is a hook at the left end of the beam).

6) Dial Gauge

The dial gauge was used to measure the deflection caused by the load. It measured in 0.01mm

4. Procedures

Unlike other labs during this one three different procedures had to be followed. Each one of them will be described separately in this section.

4.1 Procedure 1

During the first procedure the cantilever had to be clamped at its end with the edge of the bench so that the length of the beam was 500mm. Then the dial gauged must be placed underneath the beam and be calibrated so that the measurement recorded is as accurate as possible. Finally the hanger must be placed at the hook on the end of the beam and weight it with at least 8 different loads. For each load the measurement of the strain and the deflection due to the load should be recorded.

4.2 Procedure 2

The second procedure is very similar to the first one. The difference is that the cantilever is clamped further along so that the total length is 400mm and the strain gauges should not be within 50mm of the clamped section of the beam. Again here at the end of the procedure the measurements of strain and deflection have to be recorded.

4.3 Procedure 3

During the third procedure, initially the beam must be clamped so that it is 500mm long. Then apply two different loads at the end (i.e. 500g and 1000g) and measure the deflection as well as the deflection for zero loads. After that the positioned of the dial gauge must be changed by 50mm and then re-measured the value of deflection for each load as described above. This process must be repeated at least 5 times. (Note that the strain during this procedure is not necessary).

5. Results

Because the procedures followed were three and from each one the measurements were different the section of the results will be divided into three different ones.

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